Imre Juhász

Weight-based shape modification of NURBS curves

We address the problem of how to perform prescribed shape modification of NURBS curves merely by the modification of the weight of some of their control points. As is known, NURBS curves can be considered as central projections of nonrational B-spline curves. Making use of the property that there is an infinite number of nonrational B-spline curves the central projection of which is a given NURBS curve, we provide a weight-based shape modification method by means of which one can prescribe not only the new position of an arbitrary chosen point of a plane NURBS curve but the tangent direction as well.

Constrained shape modification of cubic B-spline curves by means of knots

Abstract The effect of the modification of knot values on the shape of B-spline curves is examined in this paper. The modification of a knot of a B-spline curve of order k generates a one-parameter family of curves.This family has an envelope which is also a B-spline curve with the same control polygon and of order k−1. Applying this theoretical result, three shape control methods are provided for cubic B-spline curves, that are based on the modification of three consecutive knots. The proposed methods enable local shape modifications subject to position and/or tangent constraints that can be specified within well defined limits.

A class of generalized B-spline curves

The classical B-spline functions of order k>=2 are recursively defined as a special combination of two consecutive B-spline functions of order k-1. At each step, this recursive definition is based, in general, on different reparametrizations of the strictly increasing identity (linear core) function @f(u)=u. This paper generalizes the concept of the classical normalized B-spline functions by considering monotone increasing continuously differentiable nonlinear core functions instead of the classical linear one. These nonlinear core functions are not only interesting from a theoretical perspective, but they also provide a large variety of shapes. We show that many advantageous properties (like the non-negativity, local support, the partition of unity, the effect of multiple knot values, the special case of Bernstein polynomials and endpoint interpolation conditions) of the classical normalized B-spline functions remain also valid for this generalized case, moreover we also provide characterization theorems for not so obvious (geometrical) properties like the first and higher order continuity of the generalized normalized B-spline functions, C^1 continuous envelope contact property of the family of curves obtained by altering a selected knot value between its neighboring knots. Characterization theorems are illustrated by test examples. We also outline new research directions by ending our paper with a list of open problems and conjectures underpinned by numerous successful numerical tests.

Cubic parametric curves of given tangent and curvature

We propose a constructive solution to the problem of finding a cubic parametric curve in a plane if the tangent vectors (derivatives with respect to the parameter) and signed curvatures are given at its end-points but the end-points themselves are unknown. We also show how these curves can be applied to construct blending curves subject to curvature, arc length, inflection and area constraints.

On the singularity of a class of parametric curves

We consider parametric curves that are represented by combination of control points and basis functions. We let a control point vary while the rest is held fixed. We show that the locus of the moving control point that yields a zero curvature point on the curve is a developable surface, the regression curve of which is the locus that guarantees a cusp on the curve. We also specify the surface that is described by those positions of the moving control point that yield a loop on the curve. Then we apply this approach to detect cusps, inflection points and loops of C-Bezier curves. Finally, we compare cubic Bezier, cubic rational Bezier and C-Bezier curves from singularity point of view.

Shape control of cubic B-spline and NURBS curves by knot modifications

Presents shape control methods for cubic B-spline and NURBS curves by the modification of their knot values and by the simultaneous modification of weights and knots. Theoretical aspects of knot modification are also discussed, concerning the paths of points on a curve and the existence of an envelope for the family of curves resulting from a knot modification for curves of degree k.

Modifying a knot of B-spline curves

The modification of a knot of a B-spline curve of order k generates a family of B-spline curves. We show that an envelope of this family is a B-spline curve defined by the same control polygon, and its order is k - m, where m is the multiplicity of the modified knot. Moreover, their arbitrary order derivatives differ only in a multiplier.

On interpolation by spline curves with shape parameters

Interpolation of a sequence of points by spline curves generally requires the solution of a large system of equations. In this paper we provide a method which requires only local computation instead of a global system of equations and works for a large class of curves. This is a generalization of a method which previously developed for Bspline, NURBS and trigonometric CB-spline curves. Moreover, instead of numerical shape parameters we provide intuitive, user-friendly, control point based modification of the interpolating curve and the possibility of optimization as well.

A cyclic basis for closed curve and surface modeling

We define a cyclic basis for the vectorspace of truncated Fourier series. The basis has several nice properties, such as positivity, summing to 1, that are often required in computer aided design, and that are used by designers in order to control curves by manipulating control points. Our curves have cyclic symmetry, i.e. the control points can be cyclically arranged and the curve does not change when the control points are cyclically permuted. We provide an explicit formula for the elevation of the degree from n to n+r, r>=1 and prove that the control polygon of the degree elevated curve converges to the curve itself if r tends to infinity. Variation diminishing property of the curve is also verified. The proposed basis functions are suitable for the description of closed curves and surfaces with C^~ continuity at all of their points.

Approximating the helix with rational cubic Bézier curves

The paper is on the approximation of a cylindrical helix by cubic rational Bezier curves. Two methods are given, one which guarantees exact slopes at the end-points of the approximating curve, and another which ensures 2nd-order continuity at the junction of two approximating curves. In both cases, there are error bounds which enable the helix to be approximated within any prescribed tolerance.